Editing Divergence theorem (section)

== History ==
[[Joseph-Louis Lagrange]] introduced the notion of surface integrals in 1760 and again in more general terms in 1811, in the second edition of his ''[[Mécanique analytique|Mécanique Analytique]].'' Lagrange employed surface integrals in his work on fluid mechanics.<ref name=":0">{{Cite book|last=Katz|first=Victor|title=A History of Mathematics: An Introduction|publisher=Addison-Wesley|year=2009|isbn=978-0-321-38700-4|pages=808–9|chapter=Chapter 22: Vector Analysis}}</ref> He discovered the divergence theorem in 1762.<ref>In his 1762 paper on sound, Lagrange treats a special case of the divergence theorem: Lagrange (1762) "Nouvelles recherches sur la nature et la propagation du son" (New researches on the nature and propagation of sound), ''Miscellanea Taurinensia'' (also known as: ''Mélanges de Turin'' ), '''2''': 11 – 172.  This article is reprinted as: [https://books.google.com/books?id=3TA4DeQw1NoC&pg=PA151 "Nouvelles recherches sur la nature et la propagation du son"]  in: J.A. Serret, ed., ''Oeuvres de Lagrange'', (Paris, France: Gauthier-Villars, 1867), vol. 1, pages 151–316; [https://books.google.com/books?id=3TA4DeQw1NoC&pg=PA263 on pages 263–265], Lagrange transforms triple integrals into double integrals using integration by parts.</ref>

[[Carl Friedrich Gauss]] was also using surface integrals while working on the gravitational attraction of an elliptical spheroid in 1813, when he proved special cases of the divergence theorem.<ref>C. F. Gauss (1813) [https://books.google.com/books?id=ASwoAQAAMAAJ&pg=PP355 "Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata,"] ''Commentationes societatis regiae scientiarium Gottingensis recentiores'', '''2''': 355–378; Gauss considered a special case of the theorem; see the 4th, 5th, and 6th pages of his article.</ref><ref name=":0" /> He proved additional special cases in 1833 and 1839.<ref name=":32">{{Cite journal|last=Katz|first=Victor|date=May 1979|title=A History of Stokes' Theorem|journal=Mathematics Magazine|volume=52|issue=3|pages=146–156|doi=10.1080/0025570X.1979.11976770|jstor=2690275}}</ref> But it was [[Mikhail Ostrogradsky]], who gave the first proof of the general theorem, in 1826, as part of his investigation of heat flow.<ref>Mikhail Ostragradsky presented his proof of the divergence theorem to the Paris Academy in 1826; however, his work was not published by the Academy.  He returned to St. Petersburg, Russia, where in 1828–1829 he read the work that he'd done in France, to the St. Petersburg Academy, which published his work in abbreviated form in 1831.
*His proof of the divergence theorem – "Démonstration d'un théorème du calcul intégral" (Proof of a theorem in integral calculus) – which he had read to the Paris Academy on February 13, 1826, was translated, in 1965, into Russian together with another article by him.  See:   Юшкевич А.П. (Yushkevich A.P.) and Антропова В.И. (Antropov V.I.) (1965) "Неопубликованные работы М.В. Остроградского" (Unpublished works of MV Ostrogradskii), ''Историко-математические исследования'' (Istoriko-Matematicheskie Issledovaniya / Historical-Mathematical Studies), '''16''': 49–96; see the section titled:  "Остроградский М.В. Доказательство одной теоремы интегрального исчисления" (Ostrogradskii M. V. Dokazatelstvo odnoy teoremy integralnogo ischislenia / Ostragradsky M.V.  Proof of a theorem in integral calculus).
*M. Ostrogradsky (presented:  November 5, 1828; published: 1831)  [https://books.google.com/books?id=XXMhAQAAMAAJ&pg=PA129 "Première note sur la théorie de la chaleur"] (First note on the theory of heat) ''Mémoires de l'Académie impériale des sciences de St. Pétersbourg'', series 6, '''1''': 129–133; for an abbreviated version of his proof of the divergence theorem, see pages 130–131.
*Victor J. Katz (May1979) [http://www-personal.umich.edu/~madeland/math255/files/Stokes-Katz.pdf "The history of Stokes' theorem,"] {{webarchive |url=https://web.archive.org/web/20150402154904/http://www-personal.umich.edu/~madeland/math255/files/Stokes-Katz.pdf |date=April 2, 2015 }} ''Mathematics Magazine'', '''52'''(3): 146–156;  for Ostragradsky's proof of the divergence theorem, see pages 147–148.</ref> Special cases were proven by [[George Green (mathematician)|George Green]] in 1828 in ''An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism'',<ref>George Green, ''An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism'' (Nottingham, England:  T. Wheelhouse, 1838).  A form of the "divergence theorem" appears on [https://books.google.com/books?id=GwYXAAAAYAAJ&pg=PA10 pages 10–12].</ref><ref name=":32" /> [[Siméon Denis Poisson]] in 1824 in a paper on elasticity, and [[Pierre Frédéric Sarrus|Frédéric Sarrus]] in 1828 in his work on floating bodies.<ref>Other early investigators who used some form of the divergence theorem include:
*[[Siméon Denis Poisson|Poisson]] (presented: February 2, 1824; published: 1826) [http://gallica.bnf.fr/ark:/12148/bpt6k3220m/f255.image "Mémoire sur la théorie du magnétisme"] (Memoir on the theory of magnetism), ''Mémoires de l'Académie des sciences de l'Institut de France'', '''5''': 247–338; on pages 294–296, Poisson transforms a volume integral (which is used to evaluate a quantity Q) into a surface integral.  To make this transformation, Poisson follows the same procedure that is used to prove the divergence theorem.
*[[Pierre Frédéric Sarrus|Frédéric Sarrus]] (1828) "Mémoire sur les oscillations des corps flottans" (Memoir on the oscillations of floating bodies), ''Annales de mathématiques pures et appliquées'' (Nismes), '''19''': 185–211.</ref><ref name=":32" />